Schofield Greene
12/26/2023 · Middle School
Sum, difference, and product of two functions Suppose that the functions \( f \) and \( g \) are defined for all real numbers \( x \) as follows. \[ \begin{array}{l}f(x)=2 x-4 \\ g(x)=x+6\end{array} \] Write the expressions for \( (f-g)(x) \) and \( (f \cdot g)(x) \) and evaluate \( (f+g)(4) \)
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Step-by-step Solution
Given functions \( f(x) = 2x - 4 \) and \( g(x) = x + 6 \), we need to find the expressions for \( (f-g)(x) \), \( (f \cdot g)(x) \), and evaluate \( (f+g)(4) \).
1. To find \( (f-g)(x) \), we subtract function \( g(x) \) from function \( f(x) \):
\[ (f-g)(x) = f(x) - g(x) \]
2. To find \( (f \cdot g)(x) \), we multiply function \( f(x) \) by function \( g(x) \):
\[ (f \cdot g)(x) = f(x) \cdot g(x) \]
3. To evaluate \( (f+g)(4) \), we add function \( f(x) \) and function \( g(x) \) and then substitute \( x = 4 \) into the resulting expression.
Let's calculate these expressions step by step:
1. Calculate \( (f-g)(x) \):
\[ (f-g)(x) = f(x) - g(x) = (2x - 4) - (x + 6) \]
2. Calculate \( (f \cdot g)(x) \):
\[ (f \cdot g)(x) = f(x) \cdot g(x) = (2x - 4) \cdot (x + 6) \]
3. Evaluate \( (f+g)(4) \):
\[ (f+g)(4) = f(4) + g(4) = (2 \cdot 4 - 4) + (4 + 6) \]
Let's calculate these expressions.
Simplify the expression by following steps:
- step0: Solution:
\(\left(2x-4\right)-\left(x+6\right)\)
- step1: Remove the parentheses:
\(2x-4-\left(x+6\right)\)
- step2: Remove the parentheses:
\(2x-4-x-6\)
- step3: Subtract the terms:
\(x-10\)
Calculate or simplify the expression \( (2x - 4) \cdot (x + 6) \).
Simplify the expression by following steps:
- step0: Solution:
\(\left(2x-4\right)\left(x+6\right)\)
- step1: Apply the distributive property:
\(2x\times x+2x\times 6-4x-4\times 6\)
- step2: Multiply the terms:
\(2x^{2}+12x-4x-24\)
- step3: Subtract the terms:
\(2x^{2}+8x-24\)
Calculate or simplify the expression \( (2 \cdot 4 - 4) + (4 + 6) \).
Calculate the value by following steps:
- step0: Calculate:
\(\left(2\times 4-4\right)+\left(4+6\right)\)
- step1: Multiply the numbers:
\(\left(8-4\right)+\left(4+6\right)\)
- step2: Subtract the numbers:
\(4+\left(4+6\right)\)
- step3: Add the numbers:
\(4+10\)
- step4: Add the numbers:
\(14\)
The expressions for the given functions are as follows:
1. \( (f-g)(x) = x - 10 \)
2. \( (f \cdot g)(x) = 2x^{2} + 8x - 24 \)
To evaluate \( (f+g)(4) \), we found that \( (f+g)(4) = 14 \).
Quick Answer
1. \( (f-g)(x) = x - 10 \)
2. \( (f \cdot g)(x) = 2x^{2} + 8x - 24 \)
3. \( (f+g)(4) = 14 \)
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